If \(l(B)=0\)  then we may choose \(f_1,\ldots ,f_r\)  such that the Jacobian matrix \((\partial f/\partial Y)\)  has an \(r\times r\) -minor \(M\)  such that \(M(y)\)  is invertible in \(A^{\prime }\) , that is \(M\notin q=\sigma ^{-1}(xA^{\prime })\) . By Jacobian Criterion [1]} we see that \(B_{xA^{\prime }\cap B}\cong (A[Y]/I)_q\)  is regular.
